Edge-disjoint Open Trails in Complete Bipartite Multigraphs

Let ^r K _a,b be a complete bipartite multigraph. We show a necessary and sufficient condition for a multigraph ^r K _a,b to be arbitrarily decomposable into open trails. We extend the results obtained by Balister for complete graphs (Balister in Probab Comput 10:463–499, 2001). Moreover we show that a multigraph ^r K _a,b with even r and a ≥ 3 or b ≥ 3 is arbitrarily decomposable into open and closed trails.     

On the Locally Polynomial Complexity of the Projection-Gradient Method for Solving Piecewise Quadratic Optimisation Problems

This paper proposes a method for solving optimisation problems involving piecewise quadratic functions. The method provides a solution in a finite number of iterations, and the computational complexity of the proposed method is locally polynomial of the problem dimension, i.e., if the initial point belongs to the sufficiently small neighbourhood of the solution set. Proposed method could be applied for solving large systems of linear inequalities.     

Manganese(II), cobalt(II), nickel(II), copper(II) and zinc(II) complexes with 4-oxo-4H-1-benzopyran-3-carboxaldehyde

The complexes Mn(II), Co(II), Ni(II) and Zn(II) with 4-oxo-4H-1-benzopyran-3-carboxaldehyde were synthesized and characterized by elemental analysis, infrared and UV spectroscopy, X-ray diffraction patterns, magnetic susceptibility, thermal gravimetric analysis, conductivity and also solubility measurements in water, methanol and DMF solution at 298 K. They are polycrystalline compounds with various formula and different ratio of metal ion:ligand. Their formula are following: [MnL₂(H₂O)](NO₃)₂·2H₂O, [CoL₂](NO₃)₂·3H₂O, [NiL₂](NO₃)₂·3H₂O, [CuL₂](NO₃)₂·H₂O and [ZnL₃](NO₃)₂, where L = C₁₀H₆O₃. The coordination of metal ions is through oxygen atoms present in 4-position of γ-pyrone ring and of aldehyde group of ligand. Chelates of Mn(II), Co(II), Ni(II) and Cu(II) obey Curie–Weiss law and they are high-spin complexes with the weak ligand fields. The thermal stability of analyzed complexes was studied in air at 293–1,173 K. On the basis of the thermoanalytical curves, it appears that thermal stability of anhydrous analysed chelates changed following: Cu (423 K) < Zn (438 K) ~ Co (440 K) < Ni (468 K). The gaseous products of thermal decomposition of those compounds in air atmosphere are following: CO₂, CO, NO₂, N₂O, hydrocarbons and in case of hydrates also water. The molar conductance data confirm that the all studied complexes are 1:2 electrolytes in DMF solution.     

Synthesis and structural characterization of novel 2-benzimidazolylthioureas: adducts of natural isothiocyanates and 2-amino-1-methylbenzimidazole

Adducts of natural allyl, phenethyl, and benzyl isothiocyanates and 2-amino-1-methylbenzimidazole were synthesized. After optimization of the reaction conditions, the target 2-benzimidazolylthioureas were obtained in reasonable yields. The detailed molecular and crystal structures of these compounds were characterized by spectroscopic and X-ray methods. Spectral analysis demonstrated that N-(1-methylbenzimidazolyl)-N′-allylthiourea, N-(1-methylbenzimidazolyl)-N′-benzylthiourea, and N-(1-methylbenzimidazolyl)-N′-phenethylthiourea exist in solution in an unprecedented three tautomeric forms, whose structures were corroborated unambiguously.     

meso-Alkylidene (m-benzi)pentaphyrin: a modified pentaphyrin bearing exocyclic double bonds at meso-positions

meso-Alkylidene (m-benzi)pentaphyrin containing exocyclic C=C double bonds at two meso-positions is synthesized and fully characterized for the first time. The single crystal X-ray crystallographic analysis shows a concave conformation with two pyrrole rings inverted. The first protonation occurs exclusively at core nitrogen. The synthesized compound displays concentration dependent chromogenic responses for fluoride anion in organic solvent.     

Cocrystals of 2,3,5,6‐tetrafluorobenzene‐1,4‐diol with diaza aromatic compounds

2,3,5,6‐Tetrafluorobenzene‐1,4‐diol easily forms cocrystals with heteroaromatic bases containing the pyrazine unit. In the 1:1 complexes with pyrazine, C₆H₂F₄O₂·C₄H₄N₂, (I), and quinoxaline, C₆H₂F₄O₂·C₈H₆N₂, (II), the crystal components are linked via O—H...N hydrogen bonds into one‐dimensional chains. With the largest base, phenazine, the 1:2 benzenediol–phenazine complex, C₆H₂F₄O₂·2C₁₂H₈N₂, (III), was obtained, with the molecules linked via O—H...N interactions into a discrete heterotrimer. In all three cocrystals, the two types of molecules are organized into layers via softer C—H...O and C—H...F interactions and π–π stacking interactions, with stronger hydrogen bonds linking molecules of adjacent layers. In (II) and (III), molecules are arranged into heterostacks, whereas in (I) separate stacks are formed by the heterocyclic base and the benzenediol molecule.     

Dispersive liquid–liquid microextraction applied to isolation and concentration of alkylphenols and their short-chained ethoxylates in water samples

Dispersive liquid–liquid microextraction (DLLME) coupled with high-performance liquid chromatography with fluorescence detector was applied for the determination of alkylphenols and their short-chained ethoxylates in water samples. Development of DLLME procedure included optimisation of some important parameters such as kind and volume of extracting and dispersing solvents. Under optimised conditions 50 μL of trichloroethylene in 1.5 mL of acetone were rapidly injected into 5 mL of a water sample. After centrifuging the organic phase containing the analytes was taken for evaporation with a gentle nitrogen purge and reconstituted to 50 μL of acetonitrile. The aliquot of this solution was analysed with the use of HPLC. For octylphenol (OP) and octylphenol ethoxylates (OPEOs) linearity was satisfactory in the range 8–1000 μg L⁻¹ and for nonylphenol (NP) and nonylphenol ethoxylates (NPEOs) linearity was in the range from 50 to about 3000 μg L⁻¹. Limit of quantitation was 0.1 μg L⁻¹ for OP and OPEOs and 0.3 μg L⁻¹ for NP and NPEOs. Satisfactory recoveries between 66 and 79% were obtained for environmental samples. The results showed that DLLME is a simple, rapid and sensitive analytical method for the preconcentration of trace amounts of alkylphenols and their ethoxylates in environmental water samples.     

Transversality conditions for infinite horizon variational problems on time scales

We consider problems of the calculus of variations on unbounded time scales. We prove the validity of the Euler–Lagrange equation on time scales for infinite horizon problems, and a new transversality condition.     

Multi-dimensional scalar balance laws with discontinuous flux

We consider scalar balance laws with a dissipative source term. The flux function may be discontinuous with respect to both the space variable x and the unknown quantity u. We formulate the definition of entropy weak solutions and provide existence and uniqueness to the considered problem. The problem is formulated in the framework of multi-valued mappings. The notion of entropy measure-valued solutions is used to prove the so-called contraction principle and comparison principle.